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Magazine Chapter 10: Problem 16
Find the center and the radius of the circle with the given equation. Then draw the graph. $$x^{2}+y^{2}-7 x-2 y=0$$
Short Answer
Expert verified
Center: \( (\frac{7}{2}, 1) \); Radius: \( \frac{\sqrt{53}}{2} \).
Step by step solution
01
Rewrite the Equation
Rewrite the given circle equation in the standard form by completing the square. Start with: \[x^{2} + y^{2} - 7x - 2y = 0\]Group the x and y terms: \[x^{2} - 7x + y^{2} - 2y = 0\]
02
Complete the Square
Complete the square for the x-terms and y-terms:For the x-terms: \[x^{2} - 7x\] Take half of -7, which is -\(\frac{7}{2}\), then square it to get \(\frac{49}{4}\). Add and subtract this inside the equation: \[x^{2} - 7x + \frac{49}{4} - \frac{49}{4}\]Likewise, complete the square for the y-terms: \[y^{2} - 2y\]Take half of -2, which is -1, then square it to get 1. Add and subtract this inside the equation: \[y^{2} - 2y + 1 - 1\]
03
Combine and Simplify
Combine these with the original equation and simplify: \[x^{2} - 7x + \frac{49}{4} - \frac{49}{4} + y^{2} - 2y + 1 - 1 = 0\]Rewrite the completed square terms: \[(x - \frac{7}{2})^{2} - \frac{49}{4} + (y - 1)^{2} - 1 = 0\]Combine constants on the right: \[(x - \frac{7}{2})^{2} + (y - 1)^{2} = \frac{49}{4} + 1\] \[= \frac{49}{4} + \frac{4}{4} = \frac{53}{4}\]
04
Identify the Center and Radius
The equation is now in the standard form of a circle: \( (x - h)^{2} + (y - k)^{2} = r^{2} \). From this, we see that:Center \( (h, k) \) = \( (\frac{7}{2}, 1) \)Radius \( r = \sqrt{\frac{53}{4}} = \frac{\sqrt{53}}{2}\)
05
Draw the Graph
To draw the graph:1. Plot the center at \( (\frac{7}{2}, 1) \).2. With a radius of \( \frac{\sqrt{53}}{2} \), draw a circle around this center. Measure the distance of the radius in each direction (left, right, up, and down) from the center to complete the circle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
completing the square
Completing the square is a helpful method to convert quadratic equations into a form that is easier to work with. Specifically, it transforms the equation into the format \((x - h)^2\). This makes identifying the center and radius straightforward.
For example, let's take the equation part involving x:
\(x^2 - 7x\). To complete the square, follow these steps:
For example, let's take the equation part involving x:
\(x^2 - 7x\). To complete the square, follow these steps:
- Take half of the coefficient of x, which is \(-7/2\)
- Square this half to get \( ( -7/2 )^2 = 49/4\).
- Add and subtract this squared term within the equation: \( x^2 - 7x + 49/4 - 49/4\).
standard form of a circle
The standard form of a circle is essential in coordinate geometry. The general format of this equation is: \( (x - h)^2 + (y - k)^2 = r^2\), where:
- \((h, k)\) represents the center of the circle
- \(r\) is the radius
- Center: \( (\frac{7}{2}, 1)\)
- Radius: \(\sqrt{\frac{53}{4}} = \frac{\sqrt{53}}{2}\)
graphing circles
Graphing a circle might seem complicated, but it becomes easier once you follow structured steps:
- First, identify the center of the circle, \((h, k)\).
- Next, determine the radius \(r\).
- Start plotting the center on the graph at point \((h, k)\).
- From the center, extend outward in all directions (left, right, up, and down) by the length of the radius.
- Draw the circle so all points are equidistant from the center by the radius.
coordinate geometry
Coordinate geometry is a branch of geometry that uses coordinates to represent and manipulate geometric forms. Understanding it involves knowing how equations relate to geometric figures.
For circles in coordinate geometry, the equation \((x - h)^2 + (y - k)^2 = r^2\) correlates directly with its visual representation on a plane. Each solution \((x, y)\) to the equation represents a point on the circle. It lets us graph shapes precisely and explore their properties thoroughly.
Mastering coordinate geometry enhances problem-solving skills, especially in visualizing and analyzing different shapes and their equations in math.
For circles in coordinate geometry, the equation \((x - h)^2 + (y - k)^2 = r^2\) correlates directly with its visual representation on a plane. Each solution \((x, y)\) to the equation represents a point on the circle. It lets us graph shapes precisely and explore their properties thoroughly.
Mastering coordinate geometry enhances problem-solving skills, especially in visualizing and analyzing different shapes and their equations in math.
